# Great inverted snub icosidodecahedron

Great inverted snub icosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGisid
Coxeter diagrams5/3s3s ()
Elements
Faces20+60 triangles, 12 pentagrams
Edges30+60+60
Vertices60
Vertex figureIrregular pentagon, edge lengths 1, 1, 1, 1, (5–1)/2
Measures (edge length 1)
Volume≈ 2.71387
Dihedral angles3–3: ≈ 89.78760°
5/2–3: ≈ 21.61047°
Central density13
Number of pieces780
Level of complexity65
Related polytopes
ArmyNon-uniform snub dodecahedron
RegimentGisid
DualGreat inverted pentagonal hexecontahedron
ConjugatesSnub dodecahedron, Great snub icosidodecahedron, Great inverted retrosnub icosidodecahedron
Convex coreChiral order-6 truncated pentakis dodecahedron
Abstract properties
Euler characteristic2
Topological properties
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The great inverted snub icosidodecahedron or gisid, is a uniform polyhedron. It consists of 60 snub triangles, 20 additional triangles, and 12 pentagrams. Four triangles and one pentagram meet at each vertex. It can be constructed by alternation of the great quasitruncated icosidodecahedron after setting all edge lengths to be equal.

## Measures

The circumradius R ≈ 0.64502 of the great inverted snub icosidodecahedron with unit edge length is the second to smallest positive real root of:

${\displaystyle 4096x^{12}-27648x^{10}+47104x^8-35776x^6+13872x^4-2696x^2+209.}$

Its volume V ≈ 2.71387 is given by the second to smallest positive real root of:

{\displaystyle \begin{align}&2176782336x^{12}-3195335070720x^{10}+162223191936000x^8+1030526618040000x^6\\ {} &+6152923794150000x^4-182124351550575000x^2+187445810737515625.\end{align}}

These same polynomials define the circumradii and volumes of the snub dodecahedron, the great snub icosidodecahedron, and the great inverted retrosnub icosidodecahedron.

## Related polyhedra

The great inverted disnub icosidodecahedron is a uniform polyhedron compound composed of the two opposite chiral forms of the great inverted snub icosidodecahedron.

o5/3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great stellated dodecahedron gissid {5/3,3} x5/3o3o ()
Quasitruncated great stellated dodecahedron quit gissid t{5/3,3} x5/3x3o ()
Great icosidodecahedron gid r{3,5/3} o5/3x3o ()
Truncated great icosahedron tiggy t{3,5/3} o5/3x3x ()
Great icosahedron gike {3,5/3} o5/3o3x ()
Quasirhombicosidodecahedron qrid rr{3,5/3} x5/3o3x ()
Great quasitruncated icosidodecahedron gaquatid tr{3,5/3} x5/3x3x ()
Great inverted snub icosidodecahedron gisid sr{3,5/3} s5/3s3s ()